Finite Group Databasebeta

← #28349545

Subgroups — #28349545

Order 13696.

380 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293362C211yesmaximal, characteristic, complemented
2#20293017C211yesmaximal, characteristic, complemented
3#20293014C211yesmaximal, characteristic, complemented
4#20293009C211yesmaximal, characteristic, complemented
5#20292849C211yesmaximal, characteristic, complemented
6#20292846C211yesmaximal, characteristic, complemented
7#20292842C211yesmaximal, characteristic, complemented
8#20292839C211yesmaximal, characteristic, complemented
9#20292822C211yesmaximal, characteristic, complemented
10#20292819C211yesmaximal, characteristic, complemented
11#20292758C211yesmaximal, characteristic, complemented
12#20292727C211yesmaximal, characteristic, complemented
13#20292725C211yesmaximal, characteristic, complemented
14C4×#13133803C211yesmaximal, characteristic, complemented
15#559×C107C211yesmaximal, characteristic, complemented
16C2×#13133808C2211yescharacteristic
17C2×#13133806C2211yescharacteristic
18C2×#13133806C2211yescharacteristic
19C2×#13133805C2211yescharacteristic
20C2×#13133805C2211yescharacteristic
21C2×#13133803C2211yescharacteristic
22C2×#13133803C2211yescharacteristic
23C2×#13133803C2211yescharacteristic
24C2×#13133803C2211yescharacteristic
25#16812106C2211yescharacteristic
26#16812098—21no—
27#16812097C2211yescharacteristic
28#16812097C2211yescharacteristic
29#16812094—21no—
30#16812094—21no—
31C4×Dic214C2211yescharacteristic
32C4×D428C2211yescharacteristic
33C4×D428C2211yescharacteristic
34#16812071—21no—
35#16812042C2211yescharacteristic
36C4⋊(C107⋊C8)C2211yescharacteristic
37C4⋊(C107⋊C8)C2211yescharacteristic
38C4⋊(C107⋊C8)C2211yescharacteristic
39#16812033C2211yescharacteristic
40#16812032C2211yescharacteristic
41#16812030C2211yescharacteristic
42#16812030C2211yescharacteristic
43#16812029C2211yescharacteristic
44#16812029C2211yescharacteristic
45#16812020C2211yescharacteristic
46#16812020C2211yescharacteristic
47C4×C107⋊C8C2211yescharacteristic
48#127×C107C2211yescharacteristic
49#124×C107—21no—
50#124×C107C2211yescharacteristic
51#124×C107C2211yescharacteristic
52#123×C107—21no—
53#123×C107—21no—
54C4×Q8×C107C2211yescharacteristic
55C4×Q8×C107C2211yescharacteristic
56C4×D4×C107—21no—
57C4×D4×C107C2211yescharacteristic
58C4×D4×C107C2211yescharacteristic
59C2×Dic214C2311yescharacteristic
60C2×D428—21no—
61C2×D428—21no—
62C2×D428C2311yescharacteristic
63C2×D428C2311yescharacteristic
64C2×C4×D107—21no—
65C2×C4×D107—21no—
66#13133808—41no—
67#13133806—22no—
68#13133806—41no—
69#13133805—22no—
70#13133805—41no—
71#13133803—22no—
72#13133803—41no—
73#13133803—41no—
74#13133803—41no—
75C2×C107⋊C8C2311yescharacteristic
76C2×C107⋊C8C2311yescharacteristic
77C2×C107⋊C8C2311yescharacteristic
78C2×C107⋊C8C2311yescharacteristic
79C4⋊Dic107C2311yescharacteristic
80Dic107⋊C4—21no—
81(C2×D107)⋊C4—21no—
82(C2×D107)⋊C4—21no—
83(C2×D107)⋊C4—21no—
84(C2×D107)⋊C4—21no—
85(C2×D107)⋊C4—21no—
86(C2×D107)⋊C4—21no—
87C4×Dic107—21no—
88C2×Q8×C107D411yescharacteristic
89C2×Q8×C107C2311yescharacteristic
90C2×D4×C107—21no—
91C2×D4×C107—21no—
92C2×D4×C107C2311yescharacteristic
93C2×D4×C107C2311yescharacteristic
94C22×C4×C107—21no—
95C22×C4×C107—21no—
96C4⋊C4×C107—21no—
97C4⋊C4×C107D411yescharacteristic
98C4⋊C4×C107D411yescharacteristic
99C4⋊C4×C107D411yescharacteristic
100C4⋊C4×C107C2311yescharacteristic
101C22⋊C4×C107—21no—
102C22⋊C4×C107—21no—
103C22⋊C4×C107—21no—
104C22⋊C4×C107—21no—
105C22⋊C4×C107—21no—
106C22⋊C4×C107—21no—
107C42×C107—21no—
108C42×C107C2311yescharacteristic
109C42×C107C2311yescharacteristic
110C42×C107C2311yescharacteristic
111C23×C107—21no—
112C23×C107—21no—
113C22×D107—21no—
114C22×D107—21no—
115Q8×C107—21no—
116Q8×C107—21no—
117Q8×C107—41no—
118D4×C107—21no—
119D4×C107—21no—
120D4×C107—21no—
121D4×C107—21no—
122D4×C107—42no—
123D4×C107—81no—
124C2×C4×C107—21no—
125C2×C4×C107—21no—
126C2×C4×C107—81no—
127C2×C4×C107—81no—
128C2×C4×C107C2×D411yescharacteristic
129C2×C4×C107C2×D411yescharacteristic
130C2×C4×C107C2×D411yescharacteristic
131C2×C4×C107C2×D411yescharacteristic
132C2×C4×C107C2×D411yescharacteristic
133C2×C4×C107C2×D411yescharacteristic
134C2×C4×C107C2411yescharacteristic
135C2×Dic107—21no—
136C2×Dic107—21no—
137D428—21no—
138D428—21no—
139D428—21no—
140D428—21no—
141D428—42no—
142D428—81no—
143C4×D107—81no—
144C4×D107—81no—
145Dic214—21no—
146Dic214—21no—
147C107⋊C8—21no—
148C107⋊C8—21no—
149C107⋊C8—21no—
150C107⋊C8#4112yes—
151C22×C107—41no—
152C22×C107—41no—
153C22×C107—41no—
154C22×C107—41no—
155C22×C107—41no—
156C22×C107—41no—
157C22×C107C22×D411yescharacteristic
158C2×D107—41no—
159C2×D107—41no—
160C2×D107—41no—
161C2×D107—41no—
162C2×D107—41no—
163C2×D107—41no—
164C4×C107—21no—
165C4×C107—21no—
166C4×C107—21no—
167C4×C107—21no—
168C4×C107—21no—
169C4×C107—21no—
170C4×C107—41no—
171C4×C107—41no—
172C4×C107C2×#4111yescharacteristic
173C4×C107H(5,𝔽2)11yescharacteristic
174Dic107—41no—
175Dic107—41no—
176C2×C107—81no—
177C2×C107—81no—
178C2×C107#55011yescharacteristic
179C2×C107#57511yescharacteristic
180C2×C107#57611yescharacteristic
181D107—81no—
182D107—81no—
183#3403—1071nomaximal, Sylow
184C107#340311yescharacteristic, complemented, Sylow
185#559—1071no—
186#559D10711yescharacteristic, complemented
187#496—1071no—
188#496—1071no—
189#495—1071no—
190#464—1071no—
191#464—1071no—
192#463—1071no—
193#463—1071no—
194#459—1071no—
195#459—1071no—
196#446—1071no—
197#440—1071no—
198#440—1071no—
199C4×D8—1071no—
200C2×Q16—1071no—
201C2×SD16—1071no—
202C2×SD16—1071no—
203C2×SD16—1071no—
204C2×SD16—1071no—
205C2×D8—1071no—
206C2×D8—1071no—
207C2×D8—1071no—
208C2×D8—1071no—
209#127—1071no—
210#127C2×D10711yescharacteristic
211#124—21no—
212#124—1071no—
213#124—1071no—
214#124—2141no—
215#124C2×D10711yescharacteristic
216#124C2×D10711yescharacteristic
217#123—21no—
218#123—21no—
219#123—2141no—
220#123—2141no—
221C4×Q8—1071no—
222C4×Q8C2×D10711yescharacteristic
223C4×Q8C2×D10711yescharacteristic
224C4×D4—21no—
225C4×D4—1071no—
226C4×D4—1071no—
227C4×D4—2141no—
228C4×D4C2×D10711yescharacteristic
229C4×D4C2×D10711yescharacteristic
230#107—1071no—
231C4⋊C8—1071no—
232C4⋊C8—1071no—
233C4⋊C8—1071no—
234#103—1071no—
235#103—1071no—
236#102—1071no—
237#102—1071no—
238#102—1071no—
239#102—1071no—
240#97—1071no—
241#97—1071no—
242C4×C8—1071no—
243C2×Q8—1071no—
244C2×Q8C22⋊D10711yescharacteristic
245C2×Q8C22×D10711yescharacteristic
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—1071no—
249C2×D4—1071no—
250C2×D4—2141no—
251C2×D4—2141no—
252C2×D4C22×D10711yescharacteristic
253C2×D4C22×D10711yescharacteristic
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—2141no—
257C22×C4—2141no—
258Q16—4281no—
259SD16—4281no—
260SD16—4281no—
261SD16—2142no—
262SD16—2142no—
263D8—4281no—
264D8—4281no—
265D8—4281no—
266D8—2142no—
267C2×C8—1071no—
268C2×C8—1071no—
269C2×C8—1071no—
270C2×C8—1071no—
271C4⋊C4—21no—
272C4⋊C4—1071no—
273C4⋊C4—2141no—
274C4⋊C4C22⋊D10711yescharacteristic
275C4⋊C4C22⋊D10711yescharacteristic
276C4⋊C4C22⋊D10711yescharacteristic
277C4⋊C4C22×D10711yescharacteristic
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—2141no—
285C22⋊C4—2141no—
286C22⋊C4—2141no—
287C22⋊C4—2141no—
288C22⋊C4—2141no—
289C22⋊C4—2141no—
290C42—21no—
291C42—2141no—
292C42C22×D10711yescharacteristic
293C42C22×D10711yescharacteristic
294C42C22×D10711yescharacteristic
295C23—21no—
296C23—21no—
297C23—2141no—
298C23—2141no—
299Q8—21no—
300Q8—21no—
301Q8—41no—
302Q8—2141no—
303Q8—2141no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—81no—
310D4—8561no—
311D4—4282no—
312D4—2141no—
313D4—2141no—
314D4—2141no—
315D4—2141no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—8561no—
321C2×C4—8561no—
322C2×C4—2141no—
323C2×C4—2141no—
324C2×C4C2×C22⋊D10711yescharacteristic
325C2×C4C2×C22⋊D10711yescharacteristic
326C2×C4C2×C22⋊D10711yescharacteristic
327C2×C4C2×C22⋊D10711yescharacteristic
328C2×C4C2×C22⋊D10711yescharacteristic
329C2×C4C2×C22⋊D10711yescharacteristic
330C2×C4C23×D10711yescharacteristic
331C8—1072no—
332C8—2141no—
333C8—2141no—
334C8—2141no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—4281no—
342C22—4281no—
343C22—4281no—
344C22—4281no—
345C22—4281no—
346C22—4281no—
347C22C22×C22⋊D10711yescharacteristic
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—4281no—
357C4—4281no—
358C4#41×D10711yescharacteristic
359C4#1681215511yescharacteristic
360C2—81no—
361C2—81no—
362C2—8561no—
363C2—8561no—
364C2#2029329611yescharacteristic
365C2#2029342511yescharacteristic
366C2#2029343011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#559×C107C2
socleC22×C107C22×D4
hypercenter#559D107
hypocenterC107#3403
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #559

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20293362C2semidirect
2. #20293017C2semidirect
3. #20293014C2semidirect
4. #20293009C2semidirect
5. #20292849C2semidirect
6. #20292846C2semidirect
7. #20292842C2semidirect
8. #20292839C2semidirect
9. #20292822C2semidirect
10. #20292819C2semidirect
11. #20292758C2semidirect
12. #20292727C2semidirect
13. #20292725C2semidirect
14. C4×#13133803C2semidirect
15. #559×C107C2semidirect
184. C107#3403semidirect
186. #559D107semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202933622
#202930172
#202930142
#202930092
#202928492
#202928462
#202928422
#202928392
#202928222
#202928192
#202927582
#202927272
#202927252
C4×#131338032
#559×C1072
#3403107